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Underlying structure of continuum response functions in random phase approximation
Volume 199, number 2
PHYSICS LETTERS B
17 December 1987
UNDERLYING STRUCTURE OF CONTINUUM RESPONSE FUNCTIONS
IN R A N D O M P H A S E APPROXIMATION Nguyen Van G I A I Division de Physique Th~orique ~,lnstitut de PhysiqueNucl~aire, F-91406 Orsay Cedex, France P.F. B O R T I G N O N 2 Dipartimento di Fisica, Universitgldi Padova, L35131 Padua, Italy F. Z A R D I 3 1NFN, Sezione di Padova, 1-35131 Padua, Italy and R.A. B R O G L I A Dipartimento di Fisica, Universitgtdi Milano, 1-20133Milan, Italy' and Niels Bohr lnstitute, DK-2100 Copenhagen O, Denmark Received 11 August 1987
It is shown that continuum-RPA results can be understood in terms of energy-dependent, complex RPA states. These states have finite widths which are direct escape widths. Calculations have been performed for a simple model of monopole resonance in 4°Ca,and the results are compared with those obtained by continuum-RPA and time-dependent Hartree-Fock methods.
Particle decay properties of giant resonances ( G R ) have b e c o m e a topic o f current interest since the appearance o f various coincidence experiments which measure nucleons e m i t t e d by collective multipole excitations [ 1-4]. The structure o f G R is reasonably well u n d e r s t o o d in terms o f p a r t i c l e - h o l e ( p - h ) configurations, in the f r a m e w o r k o f T a m m - D a n c o f f approximation ( T D A ) or random-phase a p p r o x i m a t i o n ( R P A ) [ 5]. F u r t h e r m o r e , the possibility o f including c o n t i n u u m effects in R P A or T D A calculations [6,7] gives information, in principle, on direct nucleon escape. In practice, things are m o r e involved because the m a i n q u a n t i t y yielded by c o n t i n u u m R P A studies is the strength d i s t r i b u t i o n function Laboratoire Associ6 au CNRS. 2 INFN, Laboratori Nazionali di Legnaro, 1-35020 Legnaro (Padua), Italy. 3 Dipartimento di Fisica, Universit/l di Padova, 1-35131 Padua, Italy.
S ( E ) corresponding to some multipole o p e r a t o r F. The function S ( E ) exhibits structures which reflect the existence o f more or less collective excitations [8], but the identification o f the widths o f these structures with direct escape widths o f G R is rather delicate a n d can be misleading if one deals with overlapping states as is generally the case. F o r instance, a recent t i m e - d e p e n d e n t H a r t r e e - F o c k ( T D H F ) study has shown that particle emission lifetimes can be somewhat different from what one can read off the strength function S ( E ) [9]. In this work, we discuss a m e t h o d to analyze the underlying structure o f the c o n t i n u u m R P A spectrum. This spectrum will be characterized by a discrete set o f complex states [ D , ) with complex eigenenergies I 2 , = c o , - i FL/2. In the f r a m e w o r k o f nuclear reaction theories using projection o p e r a t o r m e t h o d s [ 10-12] it can be shown that these states play the role o f complex d o o r w a y states through
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
155
Volume
199, number
2
PHYSICS
which the initial stage of the reaction proceeds when a GR is excited. The quantities EI, are the direct (undelayed) escape widths appearing in the scattering amplitude of reactions like A( a,a’ b)B*. Therefore, they are in principle accessible to experiment. The present study is restricted to the lp-1 h space and we thus treat only the direct escape widths of GR. A more complete scheme should contain the coupling to more complicated states and would lead to the compound (delayed) escape process as well. The complex states )D,) have been introduced in refs. [ 10,12- 141 where it was shown how they could be calculated if one uses the TDA picture. Here, we shall only recall the main steps of the method which is extended to the RPA case. First, a discrete p-h space {Q} of dimension N is built, and the hamiltonian H=H,+ I’ is diagonalized within {Q} using RPA (H, and V are the Hartree-Fock (HF) hamiltonian and residual interactions, respectively). This gives a set of 2N RPA operators {O,+ , 0,’ ; a= 1, .... N} corresponding to positive energies E, and negative energies I?, = -E,. Second, the continuum p-h space {P} orthogonal to {Q} can also be constructed. Then, the solution of H at energy E in the complete space {P} + {Q} (i.e., the continuum-RPA solution) is known if one solves in the {Qj-subspace the following effective hamiltonian [ lo]: s’f( E) = QHQ+ QHP
1 E-PHPSiq
=HN,+ W(E),
PHQ (1)
where P and Q are projection operators onto {P} and {Q}, respectively. The term W(E) is energy dependent and complex. It arises from the coupling between discrete and continuum spaces and it determines the particle decay or the eigenstates of X. The evaluation or the exact W is very complicated, and one usually approximates it by [ 121 W(E) ‘y W,(E) =QHJ
1 E-PH,P+iq
P&Q.
This approximation should be reasonable because matrix elements of QVP or PVP are expected to be small if V is short-ranged, since continuum wave functions are very small in regions where discrete wave functions are appreciable. Then, the p-h ma156
LETTERS
I7 December
b
1987
trix elements of W, can easily be calculated and complex RPA equations for the effective hamiltonian 2 derived. We found it convenient to express the complex RPA equations in the basis of real RPA states 0; 16)) 8,’ 16) instead of the usual p-h basis because this allows one to select collective real RPA states. Expanding the complex states as IDa)=DZ
lo>
R:
=&
(u/PO;
- V/yw~) 16)
one is led to the following
(3)
)
RPA-type
equations:
where d and L? are complex symmetric matrices which depend on the energy E. The solutions of (4) occur in pairs with opposite complex eigenvalues. The strength distribution S(E) of an operator F can be expressed in terms of these solutions: S(E)=-
’
:Irn 1 E-o,+iE&/2
z (bjFJD,)* a=1
- E-km,
1 -irk/2
>’
(5)
where all quantities on the RHS depend on E. We apply the above scheme to an illustrative example, namely a simple model for the isoscalar monopole resonance in 40Ca. We chose this case because it has recently been investigated in a TDHF approach [ 93 and therefore comparisons can be made. We use the same simplified Skyrme-type force, without Coulomb and spin-orbit interactions. After having calculated the HF hamiltonian Ho= Ih, in coordinate space, we diagonalize the single-particle hamiltonian h in a basis of N,= 10 harmonic oscillator states for each partial wave (l,j). We use the standard value hw = 4 1 A - 1’3MeV for the oscillator parameter. To build the {Q}-space, we keep the n lowest single-particle states for each (/,j). The results presented below have been calculated with n = 8. The RPA energy-weighted sum rule is well fulfilled within this {Q}-space. We have checked in a few cases that the results remain essentially unchanged if we use II= 10.
Volume 199, number 2
--
I
PHYSICS LETTERS B
I
1
| /
II 0.1
t
20
I
25
k I
30
E (MeV)
Fig. 1. Strength distributions in 4~Cacalculated in HF approximation (dotted curve), continuum-RPA (solid curve) and with the complex RPA states (dashed curve).
As a first test, we examine the limiting case of HF approximation where the residual interaction V is dropped. The hamiltonian H in eq. (1) must then be replaced by Ho, and approximation (2) becomes exact. We need only to work in TDA and the basis vectors O~- [o) are pure p - h configurations. The HF strength distribution S(E) for the operator r2yoo is computed according to eq. (5) (without the second term in the bracket), and the result is shown in fig. 1 (dotted curve). The exact continuum-HF distribution can be calculated directly by the method of refs. [6,7]. On the scale of fig. 1, the exact distribution and that given by eq. (5) are undistinguishable. Therefore, we can conclude that the adopted values of No and n are sufficient for an accurate description of the single-particle continuum when the coupling between {Q}- and {P}-spaces is correctly treated. We now turn to the case where the residual p-h interaction is taken into account. The full curve in fig. 1 shows the result of the exact continuum-RPA calculation. The most apparent effect of the residual interaction is to produce the narrow structure around 29.6 MeV. In the same figure, the values of S(E) deduced from the complex states JD~) are shown by the dashed curve. For each value of E, a set of states ID~) is calculated by solving eq. (4) and they are used in eq. (5). There is a good overall agreement
17 December 1987
Table 1 Energies, widths and strengths (in strength) of complex RPA states
percent of the total RPA
e). [MeV]
F]~ [MeV]
I (61FID.) I-" [%1
15.66 21.67 23.46 27.49 29.72 32.78 34.50 35.50 41.16
2.14 2.78 6.73 5.20 0.33 3.65 11.10 8.49 8.12
4.2 6.2 4.2 18.5 22.2 4.2 4.5 18.2 2.5
between the results of the two methods. Some discrepancies, for instance in the height of the narrow peak, are probably due to approximation (2). Nevertheless, the general features of the continuumRPA strength function, namely a broad bump from 20 to 29 MeV and a sharp rise at 29.6 MeV followed by a slowly decreasing tail, are well reproduced. The complex state method allows one to understand what is the underlying structure of this strength distribution. For convenience, we shall characterize each state IDa> by the quantity ] < 6 I F [ D ~ ) [2 and call it the strength of the state. In the 15-40 MeV region, there are only a few states whose strength exceeds 1% of the total RPA strength. Their energies co,~, widths P~ and strengths are listed in table 1. Each state of the table has been calculated with an energy E practically equal to cod. These states are represented in fig. 2A where the continuous strength function is also shown. There is only one strong narrow state, namely the one at co~=29.7 MeV with F~ = 330 keV. The broad bump extending from 20 to 29 MeV contains 3 states whose widths range from 2.8 MeV to 6.7 MeV. The decreasing tail above 30 MeV is produced by very wide states. From fig. 2A, one can see that the smooth parts of the strength function originate from the presence of strongly overlapping states. The difference between the narrow state and the others lies in their p-h structure. The main components of the 29.7 MeV state correspond to a particle in a bound or quasi-bound orbital and therefore its coupling to the continuum is weak, whereas broader states have important configurations where the particle is above the centrifugal barrier and can be easily emitted. 157
Volume 199, number 2
PHYSICS LETTERS B
\ 40
A
6,~-iT,/2=(610, Wo(E=E,+d,)O~ 1 6 ) .
~
~\\\\\
30 207 10g 20
J
",, ~+
e
~0
20
/
$
'
[" ~/21
O
f
10
0
"$"
i "(~1,2)
'03
r ',
[
20
! I
II
25
I, I'l
30
'll
35
!
IE (MeV)
Fig. 2. ( A ) Complex RPA states o f table 1. The widths F ~ are
indicated by horizontal lines. The strength distribution S(E) is shown in arbitrary units (dashed curve). (B) Real RPA states modified by first-order coupling to the continuum. (C) Real RPA states without coupling to the continuum. In a T D H F study o f the same m o d e l [9] it was found that a strong state near 29.6 MeV should have a width o f 240 keV while other states could be as b r o a d as 3 MeV, b u t it was not possible to be m o r e specific a b o u t the latter. The present results c o n f i r m the conclusions o f ref. [9] and give a complete dec o m p o s i t i o n o f the c o n t i n u u m - R P A strength function in terms o f complex states. All quantities related to the c o m p l e x states I D , ) are energy dependent. They can be meaningful only if they do not vary rapidly with E, or m o r e precisely if o9~, F ~ and ( 6 [ F I D ~ ) do not change m u c h when E is in an interval o f length F~. centered a r o u n d co.. One expects this to be so if the {Q}-space contains all b o u n d states a n d the m a i n part o f q u a s i - b o u n d states so that the propagator (E-PHoP+ i q ) - : o f eq. (2) has no r a p i d energy dependence. Let us denote by o 9 > - i F > / 2 a n d o 9 < - i F < / 2 the complex energies o f the state ID,) calculated with E=og,+FL/2 in eq. ( 4 ) . F o r the states o f table 1, we find Io9> co< J~F,/4. F o r the widths we obtain I F > - F < I <<,F~/3,except for the narrow state where this variation is FL/2, approximately. F o r a good description of the continuous strength function S(E), it is i m p o r t a n t to solve the RPA-like equation ( 4 ) , i.e. one cannot just treat the discretet o - c o n t i n u u m coupling W(E) perturbatively. This is illustrated by fig. 2B where the {Q}-space R P A states are drawn, displaced by the complex quantities 158
17 December 1987
(6)
The states O ,+ t 6 ) are also shown in fig. 2C for comp a r i s o n (only states having m o r e than 1% o f the total strength are d r a w n ) . The shifts 5 , and widths 7 , are too small (except for the n a r r o w state near 29.7 M e V ) , and one would have a p o o r agreement with the exact function S(E) if one uses the states calculated perturbatively. In summary, we have shown that c o n t i n u u m - R P A results can be u n d e r s t o o d in terms of complex RPA states. These states have finite widths due to the discrete-to-continuum coupling. They are energy dependent, but this dependence is sufficiently moderate to insure that their energies and widths r e m a i n meaningful quantities. As we have m e n t i o n e d before, the complex states a p p e a r in the scattering amplitude o f reactions leading to the direct decay of GR, a n d their widths are the direct escape widths. Therefore, they are an i m p o r t a n t ingredient for the analysis o f current experiments. The present work is a first step towards a m o r e detailed study of G R particle decay. One also needs to take into account the spreading o f R P A states over m o r e c o m p l i c a t e d states ( 2 p - 2 h , etc.) which will in turn decay by particle emission. We plan to apply this scheme to cases such as the isoscalar m o n o p o l e resonance in heavy nuclei for which measurements have been carried out [ 2 - 4 ] .
References [ 1] H. Ejiri, J. Phys. (Paris) 45 (1984) C4-135. [2] W. Eyrich et al., Phys. Rev. C29 (1984) 418; K. Fuchs et al., Phys. Rev. C 32 (1985) 418. [3] S. Brandenburg et al., Nucl. Phys. A 466 (1987) 29. [4] A. Bracco et al., submitted to Phys. Lett. B. [5] J. Speth and A. van der Woude, Rep. Prog. Phys. 44 (1981) 719, and references therein. [6] S. Shlomo and G.F. Bertsch, Nucl. Phys. A 243 (1975) 507. [7] K.F. Liu and Nguyen Van Giai, Phys. Lett. B 65 (1976) 23. [ 8 ] Nguyen Van Giai and H. Sagawa, Nucl. Phys. A 371 ( 1981 ) 1. [9] Ph. Chomaz, Nguyen Van Giai and S. Stringari, Phys. Lett. B 189 (1987) 375. [ 10] H. Feshbach, A.K. Kerman and R.H. Lemmer, Ann. Phys. (NY) 41 (1967) 230. [ 11 ] F. Zardi and P.F. Bortignon, Europhys. Lett. 1 (1986) 281. [ 12] S. Yoshida and S. Adachi, Z. Phys. A 325 (1986) 441. [ 13 ] S. Yoshida and S. Adachi, Nucl. Phys, A 457 (1986) 84. [ 14] S. Adachi and S. Yoshida, Nucl. Phys. A 462 (1987) 61.
PHYSICS LETTERS B
17 December 1987
UNDERLYING STRUCTURE OF CONTINUUM RESPONSE FUNCTIONS
IN R A N D O M P H A S E APPROXIMATION Nguyen Van G I A I Division de Physique Th~orique ~,lnstitut de PhysiqueNucl~aire, F-91406 Orsay Cedex, France P.F. B O R T I G N O N 2 Dipartimento di Fisica, Universitgldi Padova, L35131 Padua, Italy F. Z A R D I 3 1NFN, Sezione di Padova, 1-35131 Padua, Italy and R.A. B R O G L I A Dipartimento di Fisica, Universitgtdi Milano, 1-20133Milan, Italy' and Niels Bohr lnstitute, DK-2100 Copenhagen O, Denmark Received 11 August 1987
It is shown that continuum-RPA results can be understood in terms of energy-dependent, complex RPA states. These states have finite widths which are direct escape widths. Calculations have been performed for a simple model of monopole resonance in 4°Ca,and the results are compared with those obtained by continuum-RPA and time-dependent Hartree-Fock methods.
Particle decay properties of giant resonances ( G R ) have b e c o m e a topic o f current interest since the appearance o f various coincidence experiments which measure nucleons e m i t t e d by collective multipole excitations [ 1-4]. The structure o f G R is reasonably well u n d e r s t o o d in terms o f p a r t i c l e - h o l e ( p - h ) configurations, in the f r a m e w o r k o f T a m m - D a n c o f f approximation ( T D A ) or random-phase a p p r o x i m a t i o n ( R P A ) [ 5]. F u r t h e r m o r e , the possibility o f including c o n t i n u u m effects in R P A or T D A calculations [6,7] gives information, in principle, on direct nucleon escape. In practice, things are m o r e involved because the m a i n q u a n t i t y yielded by c o n t i n u u m R P A studies is the strength d i s t r i b u t i o n function Laboratoire Associ6 au CNRS. 2 INFN, Laboratori Nazionali di Legnaro, 1-35020 Legnaro (Padua), Italy. 3 Dipartimento di Fisica, Universit/l di Padova, 1-35131 Padua, Italy.
S ( E ) corresponding to some multipole o p e r a t o r F. The function S ( E ) exhibits structures which reflect the existence o f more or less collective excitations [8], but the identification o f the widths o f these structures with direct escape widths o f G R is rather delicate a n d can be misleading if one deals with overlapping states as is generally the case. F o r instance, a recent t i m e - d e p e n d e n t H a r t r e e - F o c k ( T D H F ) study has shown that particle emission lifetimes can be somewhat different from what one can read off the strength function S ( E ) [9]. In this work, we discuss a m e t h o d to analyze the underlying structure o f the c o n t i n u u m R P A spectrum. This spectrum will be characterized by a discrete set o f complex states [ D , ) with complex eigenenergies I 2 , = c o , - i FL/2. In the f r a m e w o r k o f nuclear reaction theories using projection o p e r a t o r m e t h o d s [ 10-12] it can be shown that these states play the role o f complex d o o r w a y states through
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
155
Volume
199, number
2
PHYSICS
which the initial stage of the reaction proceeds when a GR is excited. The quantities EI, are the direct (undelayed) escape widths appearing in the scattering amplitude of reactions like A( a,a’ b)B*. Therefore, they are in principle accessible to experiment. The present study is restricted to the lp-1 h space and we thus treat only the direct escape widths of GR. A more complete scheme should contain the coupling to more complicated states and would lead to the compound (delayed) escape process as well. The complex states )D,) have been introduced in refs. [ 10,12- 141 where it was shown how they could be calculated if one uses the TDA picture. Here, we shall only recall the main steps of the method which is extended to the RPA case. First, a discrete p-h space {Q} of dimension N is built, and the hamiltonian H=H,+ I’ is diagonalized within {Q} using RPA (H, and V are the Hartree-Fock (HF) hamiltonian and residual interactions, respectively). This gives a set of 2N RPA operators {O,+ , 0,’ ; a= 1, .... N} corresponding to positive energies E, and negative energies I?, = -E,. Second, the continuum p-h space {P} orthogonal to {Q} can also be constructed. Then, the solution of H at energy E in the complete space {P} + {Q} (i.e., the continuum-RPA solution) is known if one solves in the {Qj-subspace the following effective hamiltonian [ lo]: s’f( E) = QHQ+ QHP
1 E-PHPSiq
=HN,+ W(E),
PHQ (1)
where P and Q are projection operators onto {P} and {Q}, respectively. The term W(E) is energy dependent and complex. It arises from the coupling between discrete and continuum spaces and it determines the particle decay or the eigenstates of X. The evaluation or the exact W is very complicated, and one usually approximates it by [ 121 W(E) ‘y W,(E) =QHJ
1 E-PH,P+iq
P&Q.
This approximation should be reasonable because matrix elements of QVP or PVP are expected to be small if V is short-ranged, since continuum wave functions are very small in regions where discrete wave functions are appreciable. Then, the p-h ma156
LETTERS
I7 December
b
1987
trix elements of W, can easily be calculated and complex RPA equations for the effective hamiltonian 2 derived. We found it convenient to express the complex RPA equations in the basis of real RPA states 0; 16)) 8,’ 16) instead of the usual p-h basis because this allows one to select collective real RPA states. Expanding the complex states as IDa)=DZ
lo>
R:
=&
(u/PO;
- V/yw~) 16)
one is led to the following
(3)
)
RPA-type
equations:
where d and L? are complex symmetric matrices which depend on the energy E. The solutions of (4) occur in pairs with opposite complex eigenvalues. The strength distribution S(E) of an operator F can be expressed in terms of these solutions: S(E)=-
’
:Irn 1 E-o,+iE&/2
z (bjFJD,)* a=1
- E-km,
1 -irk/2
>’
(5)
where all quantities on the RHS depend on E. We apply the above scheme to an illustrative example, namely a simple model for the isoscalar monopole resonance in 40Ca. We chose this case because it has recently been investigated in a TDHF approach [ 93 and therefore comparisons can be made. We use the same simplified Skyrme-type force, without Coulomb and spin-orbit interactions. After having calculated the HF hamiltonian Ho= Ih, in coordinate space, we diagonalize the single-particle hamiltonian h in a basis of N,= 10 harmonic oscillator states for each partial wave (l,j). We use the standard value hw = 4 1 A - 1’3MeV for the oscillator parameter. To build the {Q}-space, we keep the n lowest single-particle states for each (/,j). The results presented below have been calculated with n = 8. The RPA energy-weighted sum rule is well fulfilled within this {Q}-space. We have checked in a few cases that the results remain essentially unchanged if we use II= 10.
Volume 199, number 2
--
I
PHYSICS LETTERS B
I
1
| /
II 0.1
t
20
I
25
k I
30
E (MeV)
Fig. 1. Strength distributions in 4~Cacalculated in HF approximation (dotted curve), continuum-RPA (solid curve) and with the complex RPA states (dashed curve).
As a first test, we examine the limiting case of HF approximation where the residual interaction V is dropped. The hamiltonian H in eq. (1) must then be replaced by Ho, and approximation (2) becomes exact. We need only to work in TDA and the basis vectors O~- [o) are pure p - h configurations. The HF strength distribution S(E) for the operator r2yoo is computed according to eq. (5) (without the second term in the bracket), and the result is shown in fig. 1 (dotted curve). The exact continuum-HF distribution can be calculated directly by the method of refs. [6,7]. On the scale of fig. 1, the exact distribution and that given by eq. (5) are undistinguishable. Therefore, we can conclude that the adopted values of No and n are sufficient for an accurate description of the single-particle continuum when the coupling between {Q}- and {P}-spaces is correctly treated. We now turn to the case where the residual p-h interaction is taken into account. The full curve in fig. 1 shows the result of the exact continuum-RPA calculation. The most apparent effect of the residual interaction is to produce the narrow structure around 29.6 MeV. In the same figure, the values of S(E) deduced from the complex states JD~) are shown by the dashed curve. For each value of E, a set of states ID~) is calculated by solving eq. (4) and they are used in eq. (5). There is a good overall agreement
17 December 1987
Table 1 Energies, widths and strengths (in strength) of complex RPA states
percent of the total RPA
e). [MeV]
F]~ [MeV]
I (61FID.) I-" [%1
15.66 21.67 23.46 27.49 29.72 32.78 34.50 35.50 41.16
2.14 2.78 6.73 5.20 0.33 3.65 11.10 8.49 8.12
4.2 6.2 4.2 18.5 22.2 4.2 4.5 18.2 2.5
between the results of the two methods. Some discrepancies, for instance in the height of the narrow peak, are probably due to approximation (2). Nevertheless, the general features of the continuumRPA strength function, namely a broad bump from 20 to 29 MeV and a sharp rise at 29.6 MeV followed by a slowly decreasing tail, are well reproduced. The complex state method allows one to understand what is the underlying structure of this strength distribution. For convenience, we shall characterize each state IDa> by the quantity ] < 6 I F [ D ~ ) [2 and call it the strength of the state. In the 15-40 MeV region, there are only a few states whose strength exceeds 1% of the total RPA strength. Their energies co,~, widths P~ and strengths are listed in table 1. Each state of the table has been calculated with an energy E practically equal to cod. These states are represented in fig. 2A where the continuous strength function is also shown. There is only one strong narrow state, namely the one at co~=29.7 MeV with F~ = 330 keV. The broad bump extending from 20 to 29 MeV contains 3 states whose widths range from 2.8 MeV to 6.7 MeV. The decreasing tail above 30 MeV is produced by very wide states. From fig. 2A, one can see that the smooth parts of the strength function originate from the presence of strongly overlapping states. The difference between the narrow state and the others lies in their p-h structure. The main components of the 29.7 MeV state correspond to a particle in a bound or quasi-bound orbital and therefore its coupling to the continuum is weak, whereas broader states have important configurations where the particle is above the centrifugal barrier and can be easily emitted. 157
Volume 199, number 2
PHYSICS LETTERS B
\ 40
A
6,~-iT,/2=(610, Wo(E=E,+d,)O~ 1 6 ) .
~
~\\\\\
30 207 10g 20
J
",, ~+
e
~0
20
/
$
'
[" ~/21
O
f
10
0
"$"
i "(~1,2)
'03
r ',
[
20
! I
II
25
I, I'l
30
'll
35
!
IE (MeV)
Fig. 2. ( A ) Complex RPA states o f table 1. The widths F ~ are
indicated by horizontal lines. The strength distribution S(E) is shown in arbitrary units (dashed curve). (B) Real RPA states modified by first-order coupling to the continuum. (C) Real RPA states without coupling to the continuum. In a T D H F study o f the same m o d e l [9] it was found that a strong state near 29.6 MeV should have a width o f 240 keV while other states could be as b r o a d as 3 MeV, b u t it was not possible to be m o r e specific a b o u t the latter. The present results c o n f i r m the conclusions o f ref. [9] and give a complete dec o m p o s i t i o n o f the c o n t i n u u m - R P A strength function in terms o f complex states. All quantities related to the c o m p l e x states I D , ) are energy dependent. They can be meaningful only if they do not vary rapidly with E, or m o r e precisely if o9~, F ~ and ( 6 [ F I D ~ ) do not change m u c h when E is in an interval o f length F~. centered a r o u n d co.. One expects this to be so if the {Q}-space contains all b o u n d states a n d the m a i n part o f q u a s i - b o u n d states so that the propagator (E-PHoP+ i q ) - : o f eq. (2) has no r a p i d energy dependence. Let us denote by o 9 > - i F > / 2 a n d o 9 < - i F < / 2 the complex energies o f the state ID,) calculated with E=og,+FL/2 in eq. ( 4 ) . F o r the states o f table 1, we find Io9> co< J~F,/4. F o r the widths we obtain I F > - F < I <<,F~/3,except for the narrow state where this variation is FL/2, approximately. F o r a good description of the continuous strength function S(E), it is i m p o r t a n t to solve the RPA-like equation ( 4 ) , i.e. one cannot just treat the discretet o - c o n t i n u u m coupling W(E) perturbatively. This is illustrated by fig. 2B where the {Q}-space R P A states are drawn, displaced by the complex quantities 158
17 December 1987
(6)
The states O ,+ t 6 ) are also shown in fig. 2C for comp a r i s o n (only states having m o r e than 1% o f the total strength are d r a w n ) . The shifts 5 , and widths 7 , are too small (except for the n a r r o w state near 29.7 M e V ) , and one would have a p o o r agreement with the exact function S(E) if one uses the states calculated perturbatively. In summary, we have shown that c o n t i n u u m - R P A results can be u n d e r s t o o d in terms of complex RPA states. These states have finite widths due to the discrete-to-continuum coupling. They are energy dependent, but this dependence is sufficiently moderate to insure that their energies and widths r e m a i n meaningful quantities. As we have m e n t i o n e d before, the complex states a p p e a r in the scattering amplitude o f reactions leading to the direct decay of GR, a n d their widths are the direct escape widths. Therefore, they are an i m p o r t a n t ingredient for the analysis o f current experiments. The present work is a first step towards a m o r e detailed study of G R particle decay. One also needs to take into account the spreading o f R P A states over m o r e c o m p l i c a t e d states ( 2 p - 2 h , etc.) which will in turn decay by particle emission. We plan to apply this scheme to cases such as the isoscalar m o n o p o l e resonance in heavy nuclei for which measurements have been carried out [ 2 - 4 ] .
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